Lesson Worksheet: Binomial Theorem: Using Partial Fractions Mathematics

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In this worksheet, we will practice decomposing rational expressions into partial fractions, then expanding them using the binomial theorem.

Q1:

By decomposing 3𝑥−1(𝑥+1)(𝑥−3) into partial fractions, find the quadratic approximation of 3𝑥−1(𝑥+1)(𝑥−3) and state the range of values of 𝑥 for which the approximation is valid.

  • A−13−119𝑥−2527𝑥,|𝑥|<1.
  • B13−119𝑥+2527𝑥, |𝑥|<3.
  • C13−119𝑥+2527𝑥, |𝑥|<1.
  • D13+119𝑥+2527𝑥, |𝑥|<1.
  • E13+119𝑥+2527𝑥, |𝑥|<3.

Q2:

By decomposing 3𝑥(1−𝑥)(𝑥+2) into partial fractions, find the cubic approximation of 3𝑥(1−𝑥)(𝑥+2) and state the range of values of 𝑥 for which the approximation is valid.

  • A32𝑥+34𝑥+98𝑥, |𝑥|<2
  • B32𝑥−34𝑥+98𝑥, |𝑥|<2
  • C−32𝑥−34𝑥−98𝑥,|𝑥|<1
  • D32𝑥−34𝑥+98𝑥, |𝑥|<1
  • E32𝑥+34𝑥+98𝑥, |𝑥|<1

Q3:

By expressing −𝑥−3(𝑥+2) in partial fractions, find the cubic expansion of −𝑥−3(𝑥+2) and state the range of values of 𝑥 for which the approximation is valid.

  • A34+12𝑥−516𝑥+316𝑥, |𝑥|<2
  • B−34+12𝑥−516𝑥+316𝑥, |𝑥|<1
  • C34+12𝑥−516𝑥+316𝑥, |𝑥|<1
  • D−34+12𝑥−516𝑥+316𝑥, |𝑥|<2
  • E−34+12𝑥−516𝑥+316𝑥, 𝑥<2

Q4:

By expressing −1+2𝑥+5𝑥(𝑥+1)(𝑥+2)(𝑥−1) in partial fractions, find the quadratic approximation of −1+2𝑥+5𝑥(𝑥+1)(𝑥+2)(𝑥−1) and state the range of values of 𝑥 for which the approximation is valid.

  • A12+114𝑥+78𝑥, |𝑥|<1
  • B12−114𝑥+78𝑥, |𝑥|<2
  • C12−114𝑥+78𝑥, |𝑥|<1
  • D12+114𝑥+78𝑥, |𝑥|<2
  • E−12+114𝑥−78𝑥, |𝑥|<1

Q5:

By writing 𝑥+3(2𝑥+1)(1−3𝑥) in the form 𝐴2𝑥+1+𝐵1−3𝑥, find the expansion of 𝑥+3(2𝑥+1)(1−3𝑥) up to the 𝑥 term and state the values of 𝑥 for which the expansion is valid.

  • A3+4𝑥+22𝑥, |𝑥|<1
  • B3−4𝑥+22𝑥, |𝑥|<12
  • C3−4𝑥+22𝑥, |𝑥|<13
  • D3+4𝑥+22𝑥, |𝑥|<13
  • E3+4𝑥+22𝑥, |𝑥|<12

Q6:

By decomposing 𝑥−𝑥−4(𝑥−2)(𝑥−3) into partial fractions, find the quadratic approximation of 𝑥−𝑥−4(𝑥−2)(𝑥−3) and state the range of values of 𝑥 for which the approximation is valid.

  • A−23−1318𝑥−35108𝑥, |𝑥|<2
  • B−53−1318𝑥−35108𝑥, |𝑥|<3
  • C−53−1318𝑥−35108𝑥, |𝑥|<2
  • D−1−1318𝑥−35108𝑥, |𝑥|<2
  • E−23−1318𝑥−35108𝑥, |𝑥|<3

Q7:

By writing 4𝑥+15𝑥−5(𝑥+3)(2𝑥−1) in the form 𝐴+𝐵𝑥+3+𝐶2𝑥−1, find the expansion of 4𝑥+15𝑥−5(𝑥+3)(2𝑥−1) up to the 𝑥 term and state the values of 𝑥 for which the expansion is valid.

  • A53−209𝑥−10627𝑥, |𝑥|<3
  • B53−209𝑥+10627𝑥, |𝑥|<12
  • C53−209𝑥−10627𝑥, |𝑥|<12
  • D−13−209𝑥−10627𝑥, |𝑥|<3
  • E−13−209𝑥−10627𝑥, |𝑥|<12

Q8:

By writing 𝑓(𝑥)=10𝑥−6𝑥+2(𝑥+1)(2𝑥−1) in the form 𝐴2𝑥−1+𝐵(2𝑥−1)+𝐶𝑥+1, find the expansion of 10𝑥−6𝑥+2(𝑥+1)(2𝑥−1) up to the 𝑥 term and state the values of 𝑥 for which the expansion is valid.

  • A2+4𝑥+10𝑥+22𝑥, |𝑥|<12
  • B2+10𝑥+22𝑥, |𝑥|<1
  • C2+4𝑥+10𝑥+22𝑥, |𝑥|<1
  • D2+10𝑥+22𝑥, |𝑥|<12
  • E1+10𝑥+22𝑥, |𝑥|<12

Find the percentage error in using approximation to estimate the value of 𝑓(0.1), giving your answer to 2 decimal places.

Q9:

By decomposing 𝑓(𝑥)=𝑥−𝑥−1(𝑥−1) into partial fractions, find the expansion of 𝑓(𝑥) up to the 𝑥 term and state the values of 𝑥 for which the expansion is valid.

  • A1+𝑥+2𝑥+3𝑥, |𝑥|<1
  • B−3𝑥−4𝑥−5𝑥, |𝑥|<1
  • C−1−𝑥−2𝑥−3𝑥, |𝑥|<1
  • D−1−3𝑥−4𝑥−5𝑥, |𝑥|<1
  • E1+3𝑥+4𝑥+5𝑥, |𝑥|<1

Find the percentage error in using approximation to estimate the value of 𝑓12.

Q10:

By decomposing 𝑓(𝑥)=𝑥+𝑥+1(𝑥+1) into partial fractions, find the expansion of 𝑓(𝑥) up to the 𝑥 term and state the values of 𝑥 for which the expansion is valid.

  • A1−4𝑥+8𝑥−13𝑥, |𝑥|<1
  • B1−4𝑥+4𝑥−7𝑥, |𝑥|<1
  • C1−2𝑥+4𝑥−7𝑥, |𝑥|<1
  • D−1+2𝑥−4𝑥+7𝑥, |𝑥|<1
  • E−1+4𝑥−8𝑥+13𝑥, |𝑥|<1

Find the percentage error in using the approximation to estimate the value of 𝑓13, giving your answer to one decimal place.

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